Question

Suppose that the roots (r_1, r_2) of the characteristic equation ((2)) are real and distinct. Then they can be written as (r_1) and (r_2) respectively if:

a) (r_1r_2 = 1)
b) (r_1 + r_2 = 2)
c) (r_1 ⋅ r_2 = 2)
d) (r_1 - r_2 = 2)

Answer 1

The question pertains to the roots of quadratic equations and their relationships. Without the coefficients of the quadratic equation, it's not possible to directly choose an option that correctly represents the general form of the roots of a quadratic equation. Matching the sum and product of the roots to provided statements would require specific equation coefficients.

Step-by-step explanation:

The question relates to the roots of a quadratic equation, which can be expressed in the form ax² + bx + c = 0. These roots, in the context of physical data, are always real and often only positive values are significant. When the roots are real and distinct, they follow the relationship given by Vieta's formulas, which state that for a quadratic equation the sum and product of the roots are r_1 + r_2 = -b/a and r_1 ⋅ r_2 = c/a, respectively. Therefore, to find which of the given expressions corresponds to these relationships, we can match product of the roots and the sum of the roots with the given options.

Given that we have four different potential relationships between the roots, without the coefficients of the quadratic equation provided, we cannot definitively choose which of the options (a, b, c, d) directly corresponds to the roots of a quadratic equation in general. However, if we were asked about a specific quadratic equation with known coefficients, we could calculate the sum and product of its roots and choose the matching statement.